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Waveguide Qed in the Squeezed Vacuum WAVEGUIDE QED IN THE SQUEEZED VACUUM A Dissertation by JIEYU YOU Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, M. Suhail Zubairy Committee Members, Alexei Sokolov Aleksei Zheltikov Philip R. Hemmer Head of Department, Grigory Rogachev May 2020 Major Subject: Physics Copyright 2020 Jieyu You ABSTRACT The well-known Purcell effect shows that the spontaneous decay rate of an emitter can be affected by the electromagnetic environment with which the emitters interact. One of the most famous and popular examples is the squeezed vacuum. Although the squeezed vacuum does not change the density of states of the electromagnetic modes, it can modify the decay rate as well as the dephasing rate of the emitters. The interaction between a single atom and the squeezed vacuum has been widely studied, while only a few publications deal with the multiple-atom system. Despite the fact that the dipole-dipole interaction induced by ordinary vacuum depends on the relative separation of atoms, there are only a few papers studying the impact of atomic separation in the squeezed vacuum. In this dissertation, we show that the interaction induced by the squeezed vacuum depends on the center of mass positions of the atoms, which is essentially different from that in the ordinary vacuum. We also illustrate how to choose the coordinate system to make the center of mass position reasonable and well-defined. Although the squeezed vacuum theory has been widely studied, it is impractical to generate a broadband squeezed vacuum reservoir which squeezes all modes in the 3-dimensional (3D) space. Recently, photon transport in a one-dimensional (1D) waveguide coupled to quantum emitters (well known as "waveguide-QED") has attracted much attention due to its possible applications in quantum device and quantum information. In contrast to the 3D case, squeezing in 1D is more experimentally feasible. Suppression of the spontaneous decay rate and the linewidth of the res- onance fluorescence atom has been experimentally demonstrated in a 1D microwave transmission line coupled to a single artificial a t om. However many-body interaction in a 1D waveguide QED system coupled to the squeezed vacuum has still not yet been studied. In this dissertation, we apply our theory to the 1D waveguide-QED system with the squeezed reservoir. Contrary to the tradi- tional result that the dephasing rate of a single atom is a constant, our calculation shows that the dephasing rate is actually position-dependent. As the dipole-dipole interaction is involved in the atomic system, both the atomic separation and center of mass position have impacts on the decay ii rate, dephasing rate, and the emitted resonance fluorescence spectrum. Moreover, the stationary maximum entangled NOON state can be achieved if atomic transition frequency is resonant with the center frequency of the squeezed vacuum. In light of the fact that two qubits can be treated as a whole to be a four level atomic system, we also study the dynamics of Ξ-type atoms driven by a squeezed vacuum. We get the interesting result that the atomic system’s steady state is a pure state, and a complete population inversion can occur when the coupling between the atomic dipole and the squeezed vacuum satisfy some certain conditions. We also mathematically prove that the steady state of a many-body system is nothing else but the direct product of that in the single atom case even when dipole-dipole interaction is involved. iii DEDICATION To my mother, Aihua Hu, and my father, Sijian You. iv ACKNOWLEDGMENTS I would like to thank my advisor, Dr. M. Suhail Zubairy, for giving me the opportunity to learn from him. He is so patient, erudite, and full of knowledge that I could not have finished this dissertation without his instruction and guidance. I also want to thank Dr. Marlan O. Scully, the leader of IQSE (The Institute for Quantum Science and Engineering) for supporting me with HEEP fellowship and inviting me to Wyoming summer school, where I could learn from many world famous physicists. Thanks also to my colleagues Dr. Zeyang Liao, Dr. Xiaodong Zeng, Dr. Shengwen Li, Dr. Phililp R. Hemmer, Dr. Girish Agarwal, and Dr. Fuli Li for their collaboration. Their help was essential in many of my projects. It has been my great pleasure to work with and learn from them. I also want to extend my gratitude to the department faculty who taught me physics courses and the staff who helped me prepare for my defense. v CONTRIBUTORS AND FUNDING SOURCES Contributors This work was supported by a dissertation committee consisting of Professor M. Suhail Zubairy, Alexei Sokolov, and Aleksei Zheltikov of the Department of Physics and Atronomy and Professor Philip R. Hemmer of the Department of Electrical & Computer Engineering. All work conducted for the dissertation was completed by the student independently. Funding Sources Graduate study was supported HEEP fellowship, a grant from the Qatar National Research Fund (QNRF) under NPRP project 8-352-1-074, and a grant from King Abdulaziz City for Science and Technology(KACST). vi NOMENCLATURE IQSE The Institute for Quantum Science and Engineering TAMU Texas A&M University QED Quantum electrodynamics 3D three dimensional TE modes traverse electric modes TM modes traverse magnetic modes vii TABLE OF CONTENTS Page ABSTRACT ......................................................................................... ii DEDICATION....................................................................................... iv ACKNOWLEDGMENTS .......................................................................... v CONTRIBUTORS AND FUNDING SOURCES ................................................. vi NOMENCLATURE ................................................................................. vii TABLE OF CONTENTS ........................................................................... viii LIST OF FIGURES ................................................................................. x 1. INTRODUCTION............................................................................... 1 2. SQUEEZED VACUUM RESERVOIR AND TRADITIONAL RESERVOIR THEORY ... 4 2.1 Introduction to the squeezed vacuum .................................................... 4 2.2 Traditional reservoir theory ............................................................... 14 3. A MODIFIED RESERVOIR THEORY IN THE SQUEEZED VACUUM................... 20 3.1 A new master equation from a modified mode function ................................ 20 3.2 Master equation in the quasi-one-dimensional waveguide .............................. 31 3.3 Dynamics of a single qubit ............................................................... 40 3.4 Dynamics of multiple qubits coupled by the dipole-dipole interaction................. 41 3.5 Generalization to multi-level atoms ...................................................... 43 4. THE STEADY STATE PROPERTIES OF ATOMS IN THE SQUEEZED VACUUM...... 50 4.1 Quantum entanglement of two qubits .................................................... 50 4.2 Resonance fluorescence of a group of atoms ............................................ 54 4.3 The steady state population inversion of a single Ξ-type atom by the squeezed vacuum 58 4.4 The steady state population inversion of multiple Ξ-type atoms by the squeezed vacuum .................................................................................... 60 5. CAVITY-CAVITY INTERACTION IN THE SQUEEZED VACUUM ...................... 69 5.1 General master equation of cavity-cavity interaction.................................... 69 5.2 Steady state of non-resonant cavities..................................................... 74 viii 5.3 Steady state of resonant cavities .......................................................... 76 6. SUMMARY AND CONCLUSIONS........................................................... 79 REFERENCES ...................................................................................... 81 ix LIST OF FIGURES FIGURE Page 2.1 The error ellipse for the squeezed vacuum. .............................................. 6 2.2 An SPDC scheme with the Type I output. ............................................... 8 2.3 The photon number distribution for the single mode squeezed vacuum. ............... 9 2.4 An SPDC scheme with the Type II output. .............................................. 10 2.5 The photon number distribution for the two-mode squeezed vacuum. ................. 12 2.6 The phase matching condition in SPDC process......................................... 15 3.1 Demonstration for squeezing all modes in a single direction. The Electromagnetic wave propagates in two opposite directions, so two pumps are needed to squeeze allmodes................................................................................... 23 3.2 (a) Schematic setup for waveguide-QED in the 1D squeezed vacuum where the vacuum is squeezed from both directions. (b) The dispersion relations inside the 1:2cπ waveguide. Here the atomic transition frequency is a , which is below the cut- off frequency of TE11 mode. Considering the fact that the atomic dipole moment is along y-axis and Ey 6= 0 only for TE10, we only need to consider TE10 mode in our calculation. ............................................................................ 32 3.3 (a) The dephasing dynamics of a single emitter in the squeezed vacuum. The black and red solid curves
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